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Theorem tpr2rico 29740
Description: For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.)
Hypotheses
Ref Expression
tpr2rico.0 𝐽 = (topGen‘ran (,))
tpr2rico.1 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))
tpr2rico.2 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
Assertion
Ref Expression
tpr2rico ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
Distinct variable groups:   𝑣,𝑢,𝑥,𝑦   𝑥,𝑟,𝐴   𝐵,𝑟   𝑥,𝐺   𝑥,𝐽   𝑥,𝑋   𝑦,𝑟,𝑋
Allowed substitution hints:   𝐴(𝑦,𝑣,𝑢)   𝐵(𝑥,𝑦,𝑣,𝑢)   𝐺(𝑦,𝑣,𝑢,𝑟)   𝐽(𝑦,𝑣,𝑢,𝑟)   𝑋(𝑣,𝑢)

Proof of Theorem tpr2rico
Dummy variables 𝑧 𝑚 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 12121 . . . . . . . . . 10 (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
21ixxf 12127 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6002 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ* → (,) Fn (ℝ* × ℝ*))
42, 3mp1i 13 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (,) Fn (ℝ* × ℝ*))
5 elssuni 4433 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 (𝐽 ×t 𝐽))
6 tpr2rico.0 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
7 retop 22475 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) ∈ Top
86, 7eqeltri 2694 . . . . . . . . . . . . . . 15 𝐽 ∈ Top
9 uniretop 22476 . . . . . . . . . . . . . . . 16 ℝ = (topGen‘ran (,))
106unieqi 4411 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
119, 10eqtr4i 2646 . . . . . . . . . . . . . . 15 ℝ = 𝐽
128, 8, 11, 11txunii 21306 . . . . . . . . . . . . . 14 (ℝ × ℝ) = (𝐽 ×t 𝐽)
135, 12syl6sseqr 3631 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 ⊆ (ℝ × ℝ))
1413ad2antrr 761 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝐴 ⊆ (ℝ × ℝ))
15 simplr 791 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋𝐴)
1614, 15sseldd 3584 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (ℝ × ℝ))
17 xp1st 7143 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
1816, 17syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ)
19 simpr 477 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ+)
2019rpred 11816 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ)
2120rehalfcld 11223 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ)
2218, 21resubcld 10402 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ)
2322rexrd 10033 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ*)
2418, 21readdcld 10013 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ)
2524rexrd 10033 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*)
26 fnovrn 6762 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((1st𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
274, 23, 25, 26syl3anc 1323 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
28 xp2nd 7144 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
2916, 28syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ)
3029, 21resubcld 10402 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ)
3130rexrd 10033 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ*)
3229, 21readdcld 10013 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ)
3332rexrd 10033 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*)
34 fnovrn 6762 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
354, 31, 33, 34syl3anc 1323 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
36 eqidd 2622 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
37 xpeq1 5088 . . . . . . . . 9 (𝑥 = (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) → (𝑥 × 𝑦) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦))
3837eqeq2d 2631 . . . . . . . 8 (𝑥 = (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) → (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦) ↔ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦)))
39 xpeq2 5089 . . . . . . . . 9 (𝑦 = (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
4039eqeq2d 2631 . . . . . . . 8 (𝑦 = (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) → (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦) ↔ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))))
4138, 40rspc2ev 3308 . . . . . . 7 (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,) ∧ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
4227, 35, 36, 41syl3anc 1323 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
43 eqid 2621 . . . . . . 7 (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) = (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
44 vex 3189 . . . . . . . 8 𝑥 ∈ V
45 vex 3189 . . . . . . . 8 𝑦 ∈ V
4644, 45xpex 6915 . . . . . . 7 (𝑥 × 𝑦) ∈ V
4743, 46elrnmpt2 6726 . . . . . 6 (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
4842, 47sylibr 224 . . . . 5 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)))
49 tpr2rico.2 . . . . 5 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
5048, 49syl6eleqr 2709 . . . 4 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
5150ralrimiva 2960 . . 3 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
52 xpss 5187 . . . . . . 7 (ℝ × ℝ) ⊆ (V × V)
5352, 16sseldi 3581 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (V × V))
5418rexrd 10033 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ*)
5519rphalfcld 11828 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ+)
5618, 55ltsubrpd 11848 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) < (1st𝑋))
5718, 55ltaddrpd 11849 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))
58 elioo1 12157 . . . . . . . . 9 ((((1st𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ↔ ((1st𝑋) ∈ ℝ* ∧ ((1st𝑋) − (𝑑 / 2)) < (1st𝑋) ∧ (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))))
5923, 25, 58syl2anc 692 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ↔ ((1st𝑋) ∈ ℝ* ∧ ((1st𝑋) − (𝑑 / 2)) < (1st𝑋) ∧ (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))))
6054, 56, 57, 59mpbir3and 1243 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))))
6129rexrd 10033 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ*)
6229, 55ltsubrpd 11848 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋))
6329, 55ltaddrpd 11849 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))
64 elioo1 12157 . . . . . . . . 9 ((((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*) → ((2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ↔ ((2nd𝑋) ∈ ℝ* ∧ ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋) ∧ (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))))
6531, 33, 64syl2anc 692 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ↔ ((2nd𝑋) ∈ ℝ* ∧ ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋) ∧ (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))))
6661, 62, 63, 65mpbir3and 1243 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))
6760, 66jca 554 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∧ (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
68 elxp7 7146 . . . . . 6 (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ↔ (𝑋 ∈ (V × V) ∧ ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∧ (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))))
6953, 67, 68sylanbrc 697 . . . . 5 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
7069ralrimiva 2960 . . . 4 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
71 mnfle 11913 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
7223, 71syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
73 pnfge 11908 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) + (𝑑 / 2)) ∈ ℝ* → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
7425, 73syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
75 mnfxr 10040 . . . . . . . . . . . . . . . . . 18 -∞ ∈ ℝ*
76 pnfxr 10036 . . . . . . . . . . . . . . . . . 18 +∞ ∈ ℝ*
77 ioossioo 12207 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ ((1st𝑋) − (𝑑 / 2)) ∧ ((1st𝑋) + (𝑑 / 2)) ≤ +∞)) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
7875, 76, 77mpanl12 717 . . . . . . . . . . . . . . . . 17 ((-∞ ≤ ((1st𝑋) − (𝑑 / 2)) ∧ ((1st𝑋) + (𝑑 / 2)) ≤ +∞) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
7972, 74, 78syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
80 ioomax 12190 . . . . . . . . . . . . . . . 16 (-∞(,)+∞) = ℝ
8179, 80syl6sseq 3630 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ ℝ)
82 mnfle 11913 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
8331, 82syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
84 pnfge 11908 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) + (𝑑 / 2)) ∈ ℝ* → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
8533, 84syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
86 ioossioo 12207 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ ((2nd𝑋) − (𝑑 / 2)) ∧ ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8775, 76, 86mpanl12 717 . . . . . . . . . . . . . . . . 17 ((-∞ ≤ ((2nd𝑋) − (𝑑 / 2)) ∧ ((2nd𝑋) + (𝑑 / 2)) ≤ +∞) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8883, 85, 87syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8988, 80syl6sseq 3630 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ ℝ)
90 xpss12 5186 . . . . . . . . . . . . . . 15 (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ ℝ ∧ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ ℝ) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
9181, 89, 90syl2anc 692 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
9291sselda 3583 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (ℝ × ℝ))
9392expcom 451 . . . . . . . . . . . 12 (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑥 ∈ (ℝ × ℝ)))
9493ancld 575 . . . . . . . . . . 11 (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ))))
9594imdistanri 726 . . . . . . . . . 10 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))))
9613adantr 481 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝐴 ⊆ (ℝ × ℝ))
97 simpr1 1065 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋𝐴)
9896, 97sseldd 3584 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ (ℝ × ℝ))
99983anassrs 1287 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑋 ∈ (ℝ × ℝ))
100 simpr 477 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑥 ∈ (ℝ × ℝ))
101 simplr 791 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ+)
102101rphalfcld 11828 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑑 / 2) ∈ ℝ+)
103 tpr2rico.1 . . . . . . . . . . . . . . 15 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))
104103cnre2csqima 29739 . . . . . . . . . . . . . 14 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑥 ∈ (ℝ × ℝ) ∧ (𝑑 / 2) ∈ ℝ+) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → ((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2))))
10599, 100, 102, 104syl3anc 1323 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → ((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2))))
106 eqid 2621 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
107103, 6, 106cnrehmeo 22660 . . . . . . . . . . . . . . . . . . . 20 𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld))
108106cnfldtopon 22496 . . . . . . . . . . . . . . . . . . . . . 22 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
109108toponunii 20647 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
11012, 109hmeof1o 21477 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
111 f1of 6094 . . . . . . . . . . . . . . . . . . . 20 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)⟶ℂ)
112107, 110, 111mp2b 10 . . . . . . . . . . . . . . . . . . 19 𝐺:(ℝ × ℝ)⟶ℂ
113112a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝐺:(ℝ × ℝ)⟶ℂ)
114113, 99ffvelrnd 6316 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑋) ∈ ℂ)
115112a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝐺:(ℝ × ℝ)⟶ℂ)
116115ffvelrnda 6315 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ ℂ)
117 sqsscirc2 29737 . . . . . . . . . . . . . . . . 17 ((((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ) ∧ 𝑑 ∈ ℝ+) → (((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2)) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑))
118114, 116, 101, 117syl21anc 1322 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2)) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑))
119118imp 445 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ ((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2))) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑)
120101rpxrd 11817 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ*)
121120adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → 𝑑 ∈ ℝ*)
122 cnxmet 22486 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) ∈ (∞Met‘ℂ)
123121, 122jctil 559 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*))
124114adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑋) ∈ ℂ)
125116adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ℂ)
126124, 125jca 554 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ))
127 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (abs ∘ − ) = (abs ∘ − )
128127cnmetdval 22484 . . . . . . . . . . . . . . . . . 18 (((𝐺𝑥) ∈ ℂ ∧ (𝐺𝑋) ∈ ℂ) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
129125, 124, 128syl2anc 692 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
130 simpr 477 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑)
131129, 130eqbrtrd 4635 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑)
132 elbl3 22107 . . . . . . . . . . . . . . . . 17 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ↔ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑))
133132biimpar 502 . . . . . . . . . . . . . . . 16 (((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) ∧ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
134123, 126, 131, 133syl21anc 1322 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
135119, 134syldan 487 . . . . . . . . . . . . . 14 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ ((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2))) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
136135ex 450 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2)) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
137105, 136syld 47 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
138 f1ocnv 6106 . . . . . . . . . . . . . . 15 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:ℂ–1-1-onto→(ℝ × ℝ))
139107, 110, 138mp2b 10 . . . . . . . . . . . . . 14 𝐺:ℂ–1-1-onto→(ℝ × ℝ)
140 f1ofun 6096 . . . . . . . . . . . . . 14 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → Fun 𝐺)
141139, 140ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐺
142 f1odm 6098 . . . . . . . . . . . . . . 15 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → dom 𝐺 = ℂ)
143139, 142ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐺 = ℂ
144116, 143syl6eleqr 2709 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ dom 𝐺)
145 funfvima 6446 . . . . . . . . . . . . 13 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ dom 𝐺) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) → (𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
146141, 144, 145sylancr 694 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) → (𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
147107, 110mp1i 13 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
148 f1ocnvfv1 6486 . . . . . . . . . . . . . . 15 ((𝐺:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
149147, 100, 148syl2anc 692 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
150149eleq1d 2683 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ↔ 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
151150biimpd 219 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
152137, 146, 1513syld 60 . . . . . . . . . . 11 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
153152imp 445 . . . . . . . . . 10 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
15495, 153syl 17 . . . . . . . . 9 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
155154ex 450 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
156155ssrdv 3589 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
157156ralrimiva 2960 . . . . . 6 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
158103mpt2fun 6715 . . . . . . . . . 10 Fun 𝐺
159158a1i 11 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → Fun 𝐺)
16013sselda 3583 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
161 f1odm 6098 . . . . . . . . . . 11 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → dom 𝐺 = (ℝ × ℝ))
162107, 110, 161mp2b 10 . . . . . . . . . 10 dom 𝐺 = (ℝ × ℝ)
163160, 162syl6eleqr 2709 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
164 simpr 477 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋𝐴)
165 funfvima 6446 . . . . . . . . . 10 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝑋𝐴 → (𝐺𝑋) ∈ (𝐺𝐴)))
166165imp 445 . . . . . . . . 9 (((Fun 𝐺𝑋 ∈ dom 𝐺) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
167159, 163, 164, 166syl21anc 1322 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
168 hmeoima 21478 . . . . . . . . . . 11 ((𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) ∧ 𝐴 ∈ (𝐽 ×t 𝐽)) → (𝐺𝐴) ∈ (TopOpen‘ℂfld))
169107, 168mpan 705 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺𝐴) ∈ (TopOpen‘ℂfld))
170106cnfldtopn 22495 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
171170elmopn2 22160 . . . . . . . . . . . 12 ((abs ∘ − ) ∈ (∞Met‘ℂ) → ((𝐺𝐴) ∈ (TopOpen‘ℂfld) ↔ ((𝐺𝐴) ⊆ ℂ ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))))
172122, 171ax-mp 5 . . . . . . . . . . 11 ((𝐺𝐴) ∈ (TopOpen‘ℂfld) ↔ ((𝐺𝐴) ⊆ ℂ ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
173172simprbi 480 . . . . . . . . . 10 ((𝐺𝐴) ∈ (TopOpen‘ℂfld) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
174169, 173syl 17 . . . . . . . . 9 (𝐴 ∈ (𝐽 ×t 𝐽) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
175174adantr 481 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
176 oveq1 6611 . . . . . . . . . . 11 (𝑚 = (𝐺𝑋) → (𝑚(ball‘(abs ∘ − ))𝑑) = ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
177176sseq1d 3611 . . . . . . . . . 10 (𝑚 = (𝐺𝑋) → ((𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
178177rexbidv 3045 . . . . . . . . 9 (𝑚 = (𝐺𝑋) → (∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
179178rspcva 3293 . . . . . . . 8 (((𝐺𝑋) ∈ (𝐺𝐴) ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
180167, 175, 179syl2anc 692 . . . . . . 7 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
181 imass2 5460 . . . . . . . . . 10 (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)))
182 f1of1 6093 . . . . . . . . . . . . 13 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)–1-1→ℂ)
183107, 110, 182mp2b 10 . . . . . . . . . . . 12 𝐺:(ℝ × ℝ)–1-1→ℂ
184 f1imacnv 6110 . . . . . . . . . . . 12 ((𝐺:(ℝ × ℝ)–1-1→ℂ ∧ 𝐴 ⊆ (ℝ × ℝ)) → (𝐺 “ (𝐺𝐴)) = 𝐴)
185183, 13, 184sylancr 694 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺 “ (𝐺𝐴)) = 𝐴)
186185sseq2d 3612 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → ((𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)) ↔ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
187181, 186syl5ib 234 . . . . . . . . 9 (𝐴 ∈ (𝐽 ×t 𝐽) → (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
188187reximdv 3010 . . . . . . . 8 (𝐴 ∈ (𝐽 ×t 𝐽) → (∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
189188adantr 481 . . . . . . 7 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → (∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
190180, 189mpd 15 . . . . . 6 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴)
191 r19.29 3065 . . . . . 6 ((∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
192157, 190, 191syl2anc 692 . . . . 5 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
193 sstr 3591 . . . . . 6 ((((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
194193reximi 3005 . . . . 5 (∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ∃𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
195192, 194syl 17 . . . 4 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
196 r19.29 3065 . . . 4 ((∀𝑑 ∈ ℝ+ 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ∃𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴) → ∃𝑑 ∈ ℝ+ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
19770, 195, 196syl2anc 692 . . 3 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
198 r19.29 3065 . . 3 ((∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ ∃𝑑 ∈ ℝ+ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
19951, 197, 198syl2anc 692 . 2 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
200 eleq2 2687 . . . . 5 (𝑟 = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (𝑋𝑟𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))))
201 sseq1 3605 . . . . 5 (𝑟 = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (𝑟𝐴 ↔ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
202200, 201anbi12d 746 . . . 4 (𝑟 = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → ((𝑋𝑟𝑟𝐴) ↔ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
203202rspcev 3295 . . 3 ((((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
204203rexlimivw 3022 . 2 (∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
205199, 204syl 17 1 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078  Fun wfun 5841   Fn wfn 5842  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  cc 9878  cr 9879  ici 9882   + caddc 9883   · cmul 9885  +∞cpnf 10015  -∞cmnf 10016  *cxr 10017   < clt 10018  cle 10019  cmin 10210   / cdiv 10628  2c2 11014  +crp 11776  (,)cioo 12117  cre 13771  cim 13772  abscabs 13908  TopOpenctopn 16003  topGenctg 16019  ∞Metcxmt 19650  ballcbl 19652  fldccnfld 19665  Topctop 20617   ×t ctx 21273  Homeochmeo 21466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-icc 12124  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cn 20941  df-cnp 20942  df-tx 21275  df-hmeo 21468  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589
This theorem is referenced by:  dya2iocnei  30125
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